Question

express the following repeated decimals as a rational number

a) 0.77777...=0.7

b) 0.757575...=0.75

 

-Find the first term and the common ration

-Find the general term, an for the given sequence

-Find a5,a10, a20

-Find sum of first n terms,Sn for the given sequence

-Find S10 and S20

a) 3,-4,(16/3,(-64/9),...

b) (-1/54),(1/81),(-2/243),...

Answer 1

(1a)

The decimal number is .

Let

Here the repeating term is digit, hence multiply by .

Subtract from

.

The rational number is .

Answer 2

(1b)

The decimal number is .

Let

Here the repeating term is digits, hence multiply by .

Subtract from .

.

The rational number is .

Answer 3

(2a)

Step 1:

The sequence is .

The first term of the sequence .

The common ratio is .

The common ratio is .

The sequence is geometric sequence.

Step 2:

Formula for term in geometric sequence is .

Substitute and .

The general term is .

Step 3:

Find and .

Substitute in .

.

Substitute in .

.

Substitute in .

.

Step 4:

The sum of first terms in a geometric series is .

Substitute and .

.

The sum of first  terms is .

Comments

Contd . . .

Step 5:

Find and .

Substitute in .

.

Substitute in .

.

Solution:

The first term is 3 and common ratio is .

The general term is .

, and .

The sum of first terms is .

, .

Answer 4

(2b)

Step 1:

The sequence is .

The first term of the sequence .

The common ratio is .

.

The common ratio is .

The sequence is geometric sequence.

Step 2:

Formula for term in geometric sequence is .

Substitute and .

The general term is .

Step 3:

Find and .

Substitute in .

.

Substitute in .

.

Substitute in .

.

Step 4:

The sum of first terms in a geometric series is .

Substitute and .

.

The sum of first terms is .

Comments

Contd . . .

Step 5:

Find and .

Substitute in .

.

Substitute in .

.

Solution:

The first term is and common ratio is .

The general term is .

, and .

The sum of first terms is .

and .